reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem
  g |^ A c= A" * g * A
proof
  let x be object;
  assume x in g |^ A;
  then consider a such that
A1: x = g |^ a and
A2: a in A by Th42;
  a" in A" by A2;
  then a" * g in A" * g by GROUP_2:28;
  hence thesis by A1,A2;
end;
